Metamath Proof Explorer

Theorem lcfrlem29

Description: Lemma for lcfr . (Contributed by NM, 9-Mar-2015)

Ref Expression
Hypotheses lcfrlem17.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
lcfrlem17.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
lcfrlem17.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
lcfrlem17.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
lcfrlem17.p ⊢ + = ( +g ‘ 𝑈 )
lcfrlem17.z ⊢ 0 = ( 0g ‘ 𝑈 )
lcfrlem17.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
lcfrlem17.a ⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
lcfrlem17.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
lcfrlem17.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
lcfrlem17.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
lcfrlem17.ne ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
lcfrlem22.b ⊢ 𝐵 = ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
lcfrlem24.t ⊢ · = ( ·𝑠 ‘ 𝑈 )
lcfrlem24.s ⊢ 𝑆 = ( Scalar ‘ 𝑈 )
lcfrlem24.q ⊢ 𝑄 = ( 0g ‘ 𝑆 )
lcfrlem24.r ⊢ 𝑅 = ( Base ‘ 𝑆 )
lcfrlem24.j ⊢ 𝐽 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) )
lcfrlem24.ib ⊢ ( 𝜑 → 𝐼 ∈ 𝐵 )
lcfrlem24.l ⊢ 𝐿 = ( LKer ‘ 𝑈 )
lcfrlem25.d ⊢ 𝐷 = ( LDual ‘ 𝑈 )
lcfrlem28.jn ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 )
lcfrlem29.i ⊢ 𝐹 = ( invr ‘ 𝑆 )
Assertion lcfrlem29 ( 𝜑 → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ∈ 𝑅 )

Proof

Step Hyp Ref Expression
1 lcfrlem17.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 lcfrlem17.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3 lcfrlem17.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 lcfrlem17.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
5 lcfrlem17.p ⊢ + = ( +g ‘ 𝑈 )
6 lcfrlem17.z ⊢ 0 = ( 0g ‘ 𝑈 )
7 lcfrlem17.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
8 lcfrlem17.a ⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
9 lcfrlem17.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 lcfrlem17.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
11 lcfrlem17.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
12 lcfrlem17.ne ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
13 lcfrlem22.b ⊢ 𝐵 = ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
14 lcfrlem24.t ⊢ · = ( ·𝑠 ‘ 𝑈 )
15 lcfrlem24.s ⊢ 𝑆 = ( Scalar ‘ 𝑈 )
16 lcfrlem24.q ⊢ 𝑄 = ( 0g ‘ 𝑆 )
17 lcfrlem24.r ⊢ 𝑅 = ( Base ‘ 𝑆 )
18 lcfrlem24.j ⊢ 𝐽 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) )
19 lcfrlem24.ib ⊢ ( 𝜑 → 𝐼 ∈ 𝐵 )
20 lcfrlem24.l ⊢ 𝐿 = ( LKer ‘ 𝑈 )
21 lcfrlem25.d ⊢ 𝐷 = ( LDual ‘ 𝑈 )
22 lcfrlem28.jn ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 )
23 lcfrlem29.i ⊢ 𝐹 = ( invr ‘ 𝑆 )
24 1 3 9 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
25 15 lmodring ⊢ ( 𝑈 ∈ LMod → 𝑆 ∈ Ring )
26 24 25 syl ⊢ ( 𝜑 → 𝑆 ∈ Ring )
27 1 3 9 dvhlvec ⊢ ( 𝜑 → 𝑈 ∈ LVec )
28 15 lvecdrng ⊢ ( 𝑈 ∈ LVec → 𝑆 ∈ DivRing )
29 27 28 syl ⊢ ( 𝜑 → 𝑆 ∈ DivRing )
30 eqid ⊢ ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
31 eqid ⊢ ( 0g ‘ 𝐷 ) = ( 0g ‘ 𝐷 )
32 eqid ⊢ { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } = { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
33 1 2 3 4 5 14 15 17 6 30 20 21 31 32 18 9 11 lcfrlem10 ⊢ ( 𝜑 → ( 𝐽 ‘ 𝑌 ) ∈ ( LFnl ‘ 𝑈 ) )
34 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
35 1 2 3 4 5 6 7 8 9 10 11 12 13 lcfrlem22 ⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
36 34 8 24 35 lsatlssel ⊢ ( 𝜑 → 𝐵 ∈ ( LSubSp ‘ 𝑈 ) )
37 4 34 lssel ⊢ ( ( 𝐵 ∈ ( LSubSp ‘ 𝑈 ) ∧ 𝐼 ∈ 𝐵 ) → 𝐼 ∈ 𝑉 )
38 36 19 37 syl2anc ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
39 15 17 4 30 lflcl ⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐽 ‘ 𝑌 ) ∈ ( LFnl ‘ 𝑈 ) ∧ 𝐼 ∈ 𝑉 ) → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 )
40 24 33 38 39 syl3anc ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 )
41 17 16 23 drnginvrcl ⊢ ( ( 𝑆 ∈ DivRing ∧ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 ∧ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 ) → ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ∈ 𝑅 )
42 29 40 22 41 syl3anc ⊢ ( 𝜑 → ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ∈ 𝑅 )
43 1 2 3 4 5 14 15 17 6 30 20 21 31 32 18 9 10 lcfrlem10 ⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ∈ ( LFnl ‘ 𝑈 ) )
44 15 17 4 30 lflcl ⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐽 ‘ 𝑋 ) ∈ ( LFnl ‘ 𝑈 ) ∧ 𝐼 ∈ 𝑉 ) → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ∈ 𝑅 )
45 24 43 38 44 syl3anc ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ∈ 𝑅 )
46 eqid ⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 )
47 17 46 ringcl ⊢ ( ( 𝑆 ∈ Ring ∧ ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ∈ 𝑅 ∧ ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ∈ 𝑅 ) → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ∈ 𝑅 )
48 26 42 45 47 syl3anc ⊢ ( 𝜑 → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ∈ 𝑅 )